A major advantage of Physics in terms of speed-of-light unit is we do NOT have to deal with huge numbers if we are to make a down to the pen and paper calculation. This is possible because speed of light is a constant even though a large number. So you can compute everything as if speed-of-light = 1 and the numbers and the related parameters such as what are known as Lorentz factors can be dealt like small numbers, especially the rapidity factor beta.
There is other definitions for rapidity factor? I think its just in terms of hyper-trigonometric functions, thats all.
For gamma you can expand the gamma in terms of beta, beta always runs from 0 to 1. Then gamma will be a binomial expansion. The binomial expansion is credited to Newton, what a man for Physics and Mathematics at the same time.
To get gamma there are 2 cases; ( — one ) you can either expand to a small number of power of beta; if the beta is small or ( — two ) if it ( — beta ) is close to 1, which is when people say ” we are dealing with an ultra-relativistic particle here ” you can do a trick. Chose a good power of beta, large enough so that you do not incur ( — more than necessary ) round off errors in your binomial expansion, which is not a mathematical problem as much as it may sound like, nonetheless it is, but take help from your Physics intuition if you are to enjoy certain privilege, you do not have to carry everything like a computer does.
This is of-course dependent on the physics problem.